study_set_12_4_07

study_set_12_4_07 - Study Set - December 4, 2007. Problem...

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Unformatted text preview: Study Set - December 4, 2007. Problem Solutions : Yates and Goodman, 6.5.1 6.5.2 6.6.2 6.7.2 7.1.4 7.2.3 7.3.1 and 7.4.3 Problem 6.5.1 Solution (a) From Table 6.1, we see that the exponential random variable X has MGF X ( s ) = - s (1) (b) Note that K is a geometric random variable identical to the geometric random variable X in Table 6.1 with parameter p = 1- q . From Table 6.1, we know that random variable K has MGF K ( s ) = (1- q ) e s 1- qe s (2) Since K is independent of each X i , V = X 1 + + X K is a random sum of random variables. From Theorem 6.12, V ( s ) = K (ln X ( s )) = (1- q ) - s 1- q - s = (1- q ) (1- q ) - s (3) We see that the MGF of V is that of an exponential random variable with parameter (1- q ) . The PDF of V is f V ( v ) = (1- q ) e- (1- q ) v v otherwise (4) Problem 6.5.2 Solution The number N of passes thrown has the Poisson PMF and MGF P N ( n ) = (30) n e- 30 /n ! n = 0 , 1 , . . . otherwise N ( s ) = e 30( e s- 1) (1) Let X i = 1 if pass i is thrown and completed and otherwise X i = 0. The PMF and MGF of each X i is P X i ( x ) = 1 / 3 x = 0 2 / 3 x = 1 otherwise X i ( s ) = 1 / 3 + (2 / 3) e s (2) The number of completed passes can be written as the random sum of random variables K = X 1 + + X N (3) 1 Since each X i is independent of N , we can use Theorem 6.12 to write K ( s ) = N (ln X ( s )) = e 30( X ( s )- 1) = e 30(2 / 3)( e s- 1) (4) We see that K has the MGF of a Poisson random variable with mean E [ K ] = 30(2 / 3) = 20, variance Var[ K ] = 20, and PMF P K ( k ) = (20) k e- 20 /k ! k = 0 , 1 , . . . otherwise (5) Problem 6.6.2 Solution Knowing that the probability that voice call occurs is 0.8 and the probability that a data call occurs is 0.2 we can define the random variable D i as the number of data calls in a single telephone call. It is obvious that for any i there are only two possible values for D i , namely 0 and 1. Furthermore for all i the D i s are independent and identically distributed withe the following PMF. P D ( d ) = . 8 d = 0 . 2 d = 1 otherwise (1) From the above we can determine that E [ D ] = 0 . 2 Var [ D ] = 0 . 2- . 04 = 0 . 16 (2) With these facts, we can answer the questions posed by the problem. (a) E [ K 100 ] = 100 E [ D ] = 20 (b) Var[ K 100 ] = p 100 Var[ D ] = 16 = 4 (c) P [ K 100 18] = 1- ( 18- 20 4 ) = 1- (- 1 / 2) = (1 / 2) = 0 . 6915 (d) P [16 K 100 24] = ( 24- 20 4 )- ( 16- 20 4 ) = (1)- (- 1) = 2(1)- 1 = 0 . 6826 Problem 6.7.2 Solution (a) Since the number of requests...
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study_set_12_4_07 - Study Set - December 4, 2007. Problem...

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